## 2185Re: [ai-geostats] Why degree of freedom is n-1

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• Aug 31, 2005
This follow-up is slighlty aside the subject line of the mailing list, but
as a geologist, this is the only statistically-flavoured one I am
subscribed to. Therefore :

Federico Pardo <federico.pardo@...> said:
> Having N samples, and then n degrees of freedom.
> One degree of freedom is used (or taken) by the mean calculation.
> Then when you calculate the variance or the standard deviation, you only
> have left n-1 degrees of freedom.

Apart a rigorous calculation I am aware of that in this very case (cf.
Peter Bossew's contribution on the same thread, that details it), gives a
proof for this rule-of-thumb, what more or less rigourous statistical
developments gives consistance to it ?

I mean, for the empirical correlation coefficient,
rhoXiYi = SUM_i=1..N( (x_i - mx).(y_i - my) / sx / sy ) / WHAT_NUMBER
Must WHAT_NUMBER be, for a kind of unbiased estimate ("a kind of" meaning
"with some eventual Fisher z-transform"...):
* N for simplicity,
* N-2 as I have most frequently seen in books that dare give this formula
(N points, minus 1 for position and 1 for dispersion ?),
* or 2N-4 -- 2N for the (x_i,y_i), minus 4 for {mx,my,sx,sy} -- as a
strict application of the rule-of-thumb seems to suggest ?

And what about, when fitting for instance a 3-parameter non-linear
function, reducing the number of degrees of freedom, to N-3 (number of
points, minus one for each function parameter ? I have never read any kind
of explanation to support it, though it seems widely

Thanks in advance for enlightments or simply tracks for other resources of
explanations.
-- Éric L.
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